Question

Suppose V is a vector space over F, dim V = n, let T be a linear transformation on V.

1. If T has an irreducible characterisctic polynomial over F, prove that {0} and V are the only T-invariant subspaces of V.

2. If the characteristic polynomial of T = g(t) h(t) for some polynomials g(t) and h(t) of degree < n , prove that V has a T-invariant subspace W such that 0 < dim W < n

Answer #1

Let V be an n-dimensional vector space. Let W and W2 be unequal
subspaces of V, each of dimension n - 1. Prove that V =W1 + W2 and
dim(Win W2) = n - 2.

Let V and W be finite-dimensional vector spaces over F, and let
φ : V → W be a linear transformation. Let dim(ker(φ)) = k, dim(V )
= n, and 0 < k < n. A basis of ker(φ), {v1, . . . , vk}, can
be extended to a basis of V , {v1, . . . , vk, vk+1, . . . , vn},
for some vectors vk+1, . . . , vn ∈ V . Prove that...

Let U and W be subspaces of a nite dimensional vector space V
such that U ∩ W = {~0}. Dene their sum U + W := {u + w | u ∈ U, w ∈
W}.
(1) Prove that U + W is a subspace of V .
(2) Let U = {u1, . . . , ur} and W = {w1, . . . , ws} be bases
of U and W respectively. Prove that U ∪ W...

Let T be a 1-1 linear transformation from a vector space V to a
vector space W. If the vectors u,
v and w are linearly independent
in V, prove that T(u), T(v),
T(w) are linearly independent in W

Let V be an n-dimensional vector space and W a vector
space that is isomorphic to V. Prove that W is also
n-dimensional. Give a clear, detailed, step-by-step
argument using the definitions of "dimension" and "isomorphic"
the Definiton of isomorphic: Let V be an
n-dimensional vector space and W a vector space that is
isomorphic to V. Prove that W is also n-dimensional. Give
a clear, detailed, step-by-step argument using the definitions of
"dimension" and "isomorphic"
The Definition of dimenion: the...

Let Mn be the vector space of all n × n matrices with real
entries. Let W = {A ∈ M3 : trace(A) = 0}, U = {B ∈ Mn : B = B t }
Verify U, W are subspaces . Find a basis for W and U and compute
dim(W) and dim(U).

Let (V, |· |v ) and (W, |· |w ) be normed vector spaces. Let T :
V → W be linear map. The kernel of T, denoted ker(T), is defined to
be the set ker(T) = {v ∈ V : T(v) = 0}. Then ker(T) is a linear
subspace of V .
Let W be a closed subspace of V with W not equal to V . Prove
that W is nowhere dense in V .

Let U be a vector space and V a subspace of U. (a) Assume dim(U)
< ∞. Show that if dim(V ) = dim(U) then V = U. (b) Assume dim(U)
= ∞ and dim(V ) = ∞. Give an example to show that it may happen
that V 6= U.

Prove that the set V of all polynomials of degree ≤ n including
the zero polynomial is vector space over the field R under usual
polynomial addition and scalar multiplication. Further, find the
basis for the space of polynomial p(x) of degree ≤ 3. Find a basis
for the subspace with p(1) = 0.

Let U and V be subspaces of the vector space W . Recall that U ∩
V is the set of all vectors ⃗v in W that are in both of U or V ,
and that U ∪ V is the set of all vectors ⃗v in W that are in at
least one of U or V
i: Prove: U ∩V is a subspace of W.
ii: Consider the statement: “U ∪ V is a subspace of W...

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